1912]
THROUGH A STRATIFIED MEDIUM, ETC.
89
The waves are now supposed to come from the positive side and are totally reflected at x — 0. The coefficient and sign of x are chosen so as to suit the formulse about to be quoted.
The solution of (97), appropriate to the present problem, is exactly the integral investigated by Airy to express the intensity of light in the neighbourhood of a caustic*. The line x = Q is, in fact, a caustic in the optical sense, being touched by all the rays. Airy's integral is
W= I cos ITT (w3 - mini) dw......................(98)
Jo
It was shown by Stokesf to satisfy (97), if
x (in his notation n) — (l-Tr)2/3 in...................(99)
Calculating by quadratures and from series proceeding by ascending powers of m, Airy tabulated W for values of in lying between in = + 5-6. For larger numerical values of m another method is necessary, for which Stokes gave the necessary formulas. Writing
where the numerical values of m and x are supposed to be taken when these quantities are negative, he found when m is positive
where E - I - -^LAiZj-il + LJLj[_l^~ 'J3 '!!_•19 ' 2^ _ (102V)
1 1.2(720)'+ 1.2.3.4(720> '"' '"( }
S = -I'5 _ 1-5-7-y.-._!?J! + (103)
When m is negative, so that W is the integral expressed by writing — m for m in (98),
The first form (101) is evidently fluctuating. The roots of W=Q are given by
(h/7r = i - 0-25 Q'028145 _ 0'026510
* being a positive integer, so that for i = 2, 3, 4, etc., we get m = 4-3631, 5-8922, 7'2436, 8'4788, etc. For i = 1, Airy's calculation gave m = 2'4955.
* Camb. Phil. Trans. 1838, Vol. vi. p. 379; 1849, Vol. vra. p. 595.
t Camb. Phil. Trans. 1850, Vol. ix.; Math, and thy8. tapers, Vol. n. p. 328.
± Here used in another sense.r which kn~ is negative. •case which best lends itself to analytical treatment is when p is a li] function of so. fc2 is then also a linear function ; and, by suitable choic the origin and scale of.*, (95) takes the form